234 Xlll INTEGRATION
9. Let (Xn)w> o be an infinite sequence of compact spaces, and let pn be a positive measure
on Xn with total mass 1, for each n.
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(a) Show that on X = XI X« there exists a unique positive measure ^ such that, for
each integer n and each finite sequence (/f)o^{<« of functions /, 6 ^R(Xi), we have
ju(/) = II jLtf(/i), where /W = fl/t(Pr^)- (Observe that the continuous functions
n
of the form n/i(pri #), f°r a^ choices of n and of f{ e ^R(Xj), form a total set in the
Banach space ^R(X).) The measure ju, is called the product of the family (/xn) and is
n
written p, = ® /xn •
«=o
(b) For each 71, let An be a /^-measurable subset of Xn. Show that the product
00 °0
A = FT An is u-measurable, and that ju(A) = Yi ^n(An).
fi = 0 n=°
(c) Let/^ 0 be a /x-integrable function on X. For each subset L of N, put L' = N — L,
and identify X with the product XL x XL>, where XL = YL x« and XL> = Yl Xn. If
n e L n e L'
x 6 X, put xL = prLjc and xL' = prL<#, and identify x with (XL,XL,). Finally, let ju,L>
be the product measure ® pa on XL', and put/L(#) = /(*L, *L') ^L- (*i/)-
n € L' »/
Now let (Ln) be an increasing sequence of subsets of N, and put g = sup fLn and
h = sup/Ln'. For each c > 0, let Ac be the set of points x e X at which g(x) > c,
and Bc the set of points x e X at which h(x) > c. Show that c • /z(Ac) <M /d/x and
c - ju(Bc) g I fdp,. (Remark that Ac is the set of points x e X for which at least one
of the/Ln(x) is >c, and express Ac as a denumerable union of mutually disjoint sets
Gn such that c • LL(G«) g f fdp,.)
JG«
(d) Suppose that (Ln) is an increasing sequence of finite subsets of N whose union is
N. Show that the/Ln tends to/almost everywhere and that/Ln tends to the constant
fdfjb almost everywhere. (For each e > 0, consider a continuous function #, depend-
ing on only finitely many variables, and such that \f—g\ dyi :ge, and apply (c)
to the function |/— g\.)
10. Let D be the discrete space {0, 1} and X the product space DN. On each factor Xn (= D)
of X let un be the measure for which ju,n(0) = /^n(l) = i, and let a = ® un be the
n e N
product measure on X (Problem 9).
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(a) For each point x = (xn)n>0 in X with xn = 0 or xn = 1, put ir(x) ~ XI ^«2"ir~I.
n = 0
Show that 77 is a continuous mapping of X onto the unit interval I — [0, 1 ] in R, and
that the image TT(IJL) of /x is Lebesgue measure on I. For each /el, the set TT"!(/)
consists of a single point, unless / is of the form k - 2"" with k integral and 0 < k < 2".
Hence the mapping /h-*/° TT induces, on passing to the quotients, an isometric
isomorphism of L£(I, A) onto L£(X, JLC) for 1 <p g + oo.
(b) For each / e J and each « ^ 1, put rn(/) — 1 — 2prn_i(7r~1(0) if t is not of the
form k * 2~m with k integral and 0 g k < 2m, and rn(0 = 0 otherwise. The function
rn is called the nth Rademacher function. These functions form a nontotal orthonormalnuous on the right, defined by w(t) — u(t)v(t) for / J> 0 and w(r) = 0 for / < 0.